Problem: Given $ \overrightarrow{BA}\perp\overrightarrow{BD}$, $ m \angle ABC = 5x - 13$, and $ m \angle CBD = 6x - 7$, find $m\angle CBD$. $B$ $A$ $D$ $C$
Solution: From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Since we are given that $\overrightarrow{BA}\perp\overrightarrow{BD}$ , we know ${m\angle ABD = 90}$ Substitute in the expressions that were given for each measure: $ {5x - 13} + {6x - 7} = {90}$ Combine like terms: $ 11x - 20 = 90$ Add $20$ to both sides: $ 11x = 110$ Divide both sides by $11$ to find $x$ $ x = 10$ Substitute $10$ for $x$ in the expression that was given for $m\angle CBD$ $ m\angle CBD = 6({10}) - 7$ Simplify: $ {m\angle CBD = 60 - 7}$ So ${m\angle CBD = 53}$.